Some Properties Of Gorenstein Objects In Triangulated Categories

Hochschild has developed relative homological algebra in categories of modules. Afterwards, Heller, Butler and Horrocks developed it in more general categories with a relative abelian structure. Its main theme consists of a selection of a class of extensions. Triangules, in triangulated categories are natural candidates for extensions. In [3], Beligiannis developed a homological algebra in C which parallels the homological algebra in an exact category in the sense of Quillen. He did this by specified a class of triangles ε∈A which is called a proper class of triangles. By fixing a proper class of triangles ε, he defined projective and injective resolutions and hence projective and injective dimensions. A version of the comparison theorem for ε-projective resolutions was presented in [3]. Based on this Beligiannis defined the ε-derived functors by using the ε-projective resolutions of objects. Naturally the ε-extension functor εxt_ε~n(—,B) was obtained.This paper continues the arguments of the Gorenstein objects in triangulated categories introduced recently by Javad Asadollahi and Shokrollah Salarian. We state the main results as follows:In section 3, we give some results on ε-ζprojective and ε-ζinjective dimensions. In section 4, we first prove a version of the comparison theorem for ε-ζprojective resolutions. Next we derive C(-, -) by ε-ζprojective resolutions in the first variable or ε-ζinjective resolutions in the second variable, and doing this, we obtain ζεxt~n(-,-) in both cases. Naturally we compare the functor ζεxt~n(-, -) and the functor ζεxt~n(-, -). Finally, we develop the the theory of 'relative homological algebra', we give some long exact complexesand a version of the Horseshoe Lemma for proper resolutions.